1 Foundations of circulation modeling systems EBS566: Estuary and Ocean Systems II Lecture 3, Winter 2010 Instructors: T. Peterson, M. Haygood, A. Baptista Division of Environmental and Biomolecular Systems, Oregon Health & Science University 2 Circulation models: what is under the hood? Correlation skill Bottom salinity 0 psu34 psu 3 Variables Velocities: u, v, w Water level: Water density: Salinity and temperatures: S, T Basic variables: 7 Equations needed: 7 4 Process Code & computer Differential Equations + Boundary conditions + Topology/ bathymetry Numerical algorithm Algebraic equations Solution (variables are known at grid locations) Discretization (grid) Post-processing Skill assessment 5 Topology/bathymetry 6 Discretization (space) Refined grid (hi-res) # nodes: # elements: # levels 24 min element area: 942 m^2 max element area: m^2 fDB16 7 The consequences of grid resolution Time step consideration? 8 Grays River: example of cascading grids 9 Grays river: detail 10 Introduction to governing equations Continuity Salt and heat conservation Depth-averaged form: 11 Introduction to governing equations Conservation of momentum (from Newtons 2 nd law: f=ma) 12 Introduction to governing equations Equation of state = (s, T, p) Turbulence closure equations 13 Consider a control volume of infinitesimal size dx dy dz Mass inside volume = Let density = Let velocity = Mass flux into the control volume = Mass flux out of the control volume = Conservation of mass - water 14 Conservation of mass states that Rate of change of mass inside the system = Mass flux into of the system Mass flux out the system Thus Conservation of mass-water and, after differentiation by parts 15 Rearranging, For incompressible fluids, like water and, thus Conservation of mass - water 16 Consider a 1D system with stationary fluid and a solute that is diffusing Let flux of mass per unit area entering the system = Let flux of mass per unit area leaving the system = Let concentration (mass /unit volume) of solute inside the control volume = C Conservation of mass of a solute dx dy dz 17 Conservation of mass states that Mass flux of solute leaving the system mass flux of solute entering the system = rate of change of solute in the system Thus or Conservation of mass of a solute - diffusion or 18 How do we quantify q D ? In a static fluid, flux of concentration (q), occurs due to random molecular motion It is not feasible to reproduce molecular motion on a large scale. Thus, we wish to represent the molecular motion by the macroscpoic property of the solute (its concentration, C) Also, from observation we know In a fluid of constant C (well mixed liquid), there is no net flux of concentration Solute moves from a region of high concentration to regions of low concentration Over some finite time scale, the solute does not show any preferential direction of motion Conservation of mass of a solute - diffusion 19 Based on these observations, Adolph Fick (1855) hypothesized that or in three dimensions Ficks law Diffusion coefficient Applying Ficks law to the 1D mass conservation equation for a solute, we get or (molecular processes are represented by an empirical coefficient analogous to diffusivity) Conservation of mass of a solute - diffusion